The pentagon shape

A pentagon is a flat shape with five straight sides, five corners (vertices) and five angles. The name comes from the Greek pente, meaning five. Every pentagon shape has interior angles that add up to 540°. In a regular pentagon, all five sides are the same length and every angle is 108°.

What is a pentagon?

A pentagon is a polygon: a closed 2D shape made only of straight sides that don't cross each other. With five sides, it comes after the quadrilateral and before the hexagon. A simple house drawing, a square with a triangle on top for the roof, is a pentagon: count the sides and there are five.

In England, AQA's GCSE Maths specification expects students to know and use the names pentagon, hexagon, octagon and decagon.

Types of pentagon

• Regular pentagon: all five sides equal and all five angles equal, at 108° each.
• Irregular pentagon: any pentagon that is not regular. The sides or angles, or both, are different.
• Convex pentagon: every angle is less than 180°, so all the corners point outwards. A regular pentagon is convex.
• Concave pentagon: at least one angle is more than 180° (a reflex angle), so one corner points inwards, like an arrowhead with a notch.
• Equilateral pentagon: all sides equal but the angles can differ. Squash a regular pentagon and you get one.

Every type, regular or not, convex or concave, still has five sides and angles that add up to 540°.

Pentagon angles

Why the angles add up to 540°

Pick one corner and draw straight lines to the two corners that aren't next to it. That splits the pentagon into three triangles. Each triangle's angles add up to 180°, and together they make up exactly the pentagon's angles:

The same trick works for any polygon. A shape with n sides splits into n − 2 triangles, so its interior angles add up to (n − 2) × 180°. For a pentagon, (5 − 2) × 180° = 540°.

Angles of a regular pentagon

A regular pentagon shares the 540° equally between five angles:

The exterior angle is the turn you make at each corner if you walk round the edge. Walk all the way round any polygon and you turn through one full turn, 360°, so each of the five equal turns in a regular pentagon is 360° ÷ 5 = 72°. Interior and exterior angles sit on a straight line, and 108° + 72° = 180°, which checks out.

Properties of a regular pentagon

Draw all five diagonals of a regular pentagon and they make a five-pointed star called a pentagram, with a smaller upside-down pentagon in the middle. Each diagonal is about 1.618 times as long as a side. That number, (1 + √5) ÷ 2, is the golden ratio.

Perimeter and area of a pentagon

Perimeter

The perimeter is the distance round the outside, so add the five sides. For a regular pentagon with side s, that is 5s. A regular pentagon with 6 cm sides has a perimeter of 30 cm, and one with a perimeter of 42 cm has sides of 42 ÷ 5 = 8.4 cm.

Area of a regular pentagon

There are two common ways. The first uses the apothem a, the distance from the centre (center in the US) to the middle of a side:

This works because the pentagon splits into five identical triangles meeting at the centre, each with base s and height a. The second way uses only the side length:

Example: a regular pentagon with 6 cm sides

Find the area of a regular pentagon with sides of 6 cm.

Using the side formula: 1.720 × 6² = 1.720 × 36 ≈ 61.9 cm².

Using the apothem: each of the five centre triangles has an angle of 360° ÷ 5 = 72° at the centre. Cut one in half to get a right-angled triangle with a 36° angle, an opposite side of 3 cm and the apothem as the adjacent side. So:

Then the area is ½ × 30 × 4.13 ≈ 61.9 cm², the same answer. The second method uses SOH CAH TOA, which the Trigonometry course teaches from scratch, and you can check the triangle in the trigonometry calculator.

Area of an irregular pentagon

There is no single formula. Split the shape into pieces you can find the area of, usually rectangles and triangles, and add them. For the house shape, that is the area of the square plus the area of the triangle roof. For a pentagon on a grid, you can also draw a rectangle round it and subtract the corner triangles.

Worked examples on pentagon angles

Example 1: the house shape

A pentagon has three right angles, and the other two angles are equal. Find them.

The three right angles make 270°. That leaves 540 − 270 = 270° for the other two, so each is 270 ÷ 2 = 135°. This is the house shape with a roof whose top corner is a right angle.

Example 2: angles in algebra

The angles of a pentagon are 2x, 3x, 4x, 4x and 5x. Find the largest angle.

The largest angle is 5x = 5 × 30 = 150°. The five angles are 60°, 90°, 120°, 120° and 150°, which add up to 540°. Always check the total: it catches slips in the algebra.

Example 3: is it regular?

A pentagon has five equal sides and one angle of 100°. Is it regular? No. A regular pentagon needs every angle to be 108°, so equal sides alone are not enough. This one is an equilateral pentagon.

Common mistakes

• Using 360° for the angle sum. That is a quadrilateral. A pentagon's interior angles add up to 540°.
• Mixing up interior and exterior angles. In a regular pentagon, the inside angle is 108° and the turn at each corner is 72°.
• Calling every five-sided shape regular. Regular means equal sides and equal angles.
• Splitting into the wrong number of triangles. From one corner you get three triangles, not five. Five triangles only works from a point in the middle, and then you must subtract the 360° around that point.

To practise polygon angles among other topics in a short timed paper, try today's Pre-Algebra & Algebra 1 or Geometry, Algebra 2 & SAT Math paper. In the UK, the same practice opens as GCSE Maths Foundation.

Read next

Maths · GeometryAngles: types, angle facts and how to find themWhat angles are, the types of angles from acute to reflex, how to measure them, every angle fact you need, and how to find missing angles, with examples.